An eradication agent acting on a binary cell population model: mathematical analysis

Abstract

In this paper we study a system of partial differential equations that couple nonfickian diffusion of one of two species with the fickian diffusion of a chemical or biological agent. This system can be used to describe the evolution a population (biological species, cells) that switch between two phenotypes: migratory and proliferative. This switching process is very common in certain types of cancer or even viral diseases. It is assumed that the population/cell migration occurs in a viscoelastic extracellular matrix (ECM). This has an important role in the transport inducing to a nonfickian motion. A chemical agent, which can represent, for instance, a treatment drug in a cancer disease scenario, acts on the population according to Michaelis-Menten or linear reactions, eliminating individuals from the population. Under suitable regularity conditions we establish upper bounds for an energy functional, with respect to the L2 norm, leading to the stability of the model. Agent administration protocols based on stronger assumptions on the agent/population kinetics are also proposed with the purpose of controlling the total amount of individuals of the population. These protocols are based on suitable estimates for the total mass of individuals in the system after a certain number of agent administration sessions. A numerical method based on finite differences and finite elements is introduced and its stability properties are analysed. The qualitative behaviour of the solutions of the system is explored and discussed. Several protocols are defined within the context of chemotherapy in brain tumours and their effectiveness is numerically studied considering Michaelis-Menten or linear chemical effects.